In general, only the former is locally constant for a log flat morphism. However, if \(f\)  is \(\mathbb {Q}\) -integral, as defined in [1]} I.4.7.4, then by [2]} Proposition \(A.7(2)\) , the two are equal in an étale neighborhood of \(x\) . Moreover, the following definition and lemma indicate how we may produce such morphisms, starting from a morphism \(X \rightarrow Y\)  which is only log smooth:
